$ A = \left[\begin{array}{r}2 \\ -1\end{array}\right]$ $ F = \left[\begin{array}{rr}1 & 4\end{array}\right]$ Is $ A F$ defined?
Answer: In order for multiplication of two matrices to be defined, the two inner dimensions must be equal. If the two matrices have dimensions $( m \times  n)$ and $( p \times q)$ , then $ n$ (number of columns in the first matrix) must equal $ p$ (number of rows in the second matrix) for their product to be defined. How many columns does the first matrix, $ A$ , have? How many rows does the second matrix, $ F$ , have? Since $ A$ has the same number of columns (1) as $ F$ has rows (1), $ A F$ is defined.